Agree. Faced same condition before. As experienced earlier few things might help you at some extend.
You need to convey them the pros & cons of the automation testing.
You need to give them a brief about the use of automation as well. I mean with lesser knowledge one can expect to much from you. You need to take initiatives to keep balance between actual possible things to automate and expectations
As per my knowledge, In your case of browser automation we can ensure that everything is working well 24*7. This is a primary goal of browser automation.
Communicate clearly if we are not bothering about things should work ALWAYS then we should think of reduce the dependency & efforts on browser automation
Limitations: Its really out of scope [as of now at lease] to automate browser to catch images or content overlapping, veriication of image quality and visibility, crosschecking on different versions of browser in [responsive] website